Vector autoregression (VAR) is a stochastic process model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregressive model (AR model) by allowing for more than one evolving variable. All variables in a VAR enter the model in the same way: each variable has an equation explaining its evolution based on its own lagged values, the lagged values of the other model variables, and an error term. VAR modeling does not require as much knowledge about the forces influencing a variable as do structural models with simultaneous equations: The only prior knowledge required is a list of variables which can be hypothesized to affect each other intertemporally.

A VAR model describes the evolution of a set of k variables (called endogenous variables) over the same sample period (t = 1, ..., T) as a linear function of only their past values. The variables are collected in a k × 1 vectoryt, which has as the i th element, yi,t, the observation at time "t" of the i th variable. For example, if the i th variable is GDP, then yi,t is the value of GDP at time t.

where the l-periods back observation yt−l is called the l-th lag of y, c is a k × 1 vector of constants (intercepts), Ai is a time-invariant k × kmatrix and et is a k × 1 vector of error terms satisfying

E(etet−k′)=0{\displaystyle \mathrm {E} (e_{t}e_{t-k}')=0\,} for any non-zero k — there is no correlation across time; in particular, no serial correlation in individual error terms.[1]

A pth-order VAR is also called a VAR with p lags. The process of choosing the maximum lag p in the VAR model requires special attention because inference is dependent on correctness of the selected lag order.[2][3]

A VAR with p lags can always be equivalently rewritten as a VAR with only one lag by appropriately redefining the dependent variable. The transformation amounts to stacking the lags of the VAR(p) variable in the new VAR(1) dependent variable and appending identities to complete the number of equations.

where c0 is a k × 1 vector of constants, Bi is a k × k matrix (for every i = 0, ..., p) and εt is a k × 1 vector of error terms. The main diagonal terms of the B0 matrix (the coefficients on the ith variable in the ith equation) are scaled to 1.

The error terms εt (structural shocks) satisfy the conditions (1) - (3) in the definition above, with the particularity that all the elements off the main diagonal of the covariance matrix E(ϵtϵt′)=Σ{\displaystyle \mathrm {E} (\epsilon _{t}\epsilon _{t}')=\Sigma } are zero. That is, the structural shocks are uncorrelated.

Note that y2,t can have a contemporaneous effect on y1,t if B0;1,2 is not zero. This is different from the case when B0 is the identity matrix (all off-diagonal elements are zero — the case in the initial definition), when y2,t can impact directly y1,t+1 and subsequent future values, but not y1,t.

From an economic point of view, if the joint dynamics of a set of variables can be represented by a VAR model, then the structural form is a depiction of the underlying, "structural", economic relationships. Two features of the structural form make it the preferred candidate to represent the underlying relations:

1. Error terms are not correlated. The structural, economic shocks which drive the dynamics of the economic variables are assumed to be independent, which implies zero correlation between error terms as a desired property. This is helpful for separating out the effects of economically unrelated influences in the VAR. For instance, there is no reason why an oil price shock (as an example of a supply shock) should be related to a shift in consumers' preferences towards a style of clothing (as an example of a demand shock); therefore one would expect these factors to be statistically independent.

2. Variables can have a contemporaneous impact on other variables. This is a desirable feature especially when using low frequency data. For example, an indirect tax rate increase would not affect tax revenues the day the decision is announced, but one could find an effect in that quarter's data.

Note that in the reduced form all right hand side variables are predetermined at time t. As there are no time t endogenous variables on the right hand side, no variable has a direct contemporaneous effect on other variables in the model.

However, the error terms in the reduced VAR are composites of the structural shocks et = B0−1εt. Thus, the occurrence of one structural shock εi,t can potentially lead to the occurrence of shocks in all error terms ej,t, thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR

Vector autoregression models often involve the estimation of many parameters. For example, with seven variables and four lags, each matrix of coefficients for a given lag length is 7 by 7, and the vector of constants has 7 elements, so a total of 49×4 + 7 = 203 parameters are estimated, substantially lowering the degrees of freedom of the regression (the number of data points minus the number of parameters to be estimated). This can hurt the accuracy of the parameter estimates and hence of the forecasts given by the model.

Consider the first-order case (i.e., with only one lag), with equation of evolution

yt=Ayt−1+et,{\displaystyle y_{t}=Ay_{t-1}+e_{t},}

for evolving (state) vector y{\displaystyle y} and vector e{\displaystyle e} of shocks. To find, say, the effect of the j-th element of the vector of shocks upon the i-th element of the state vector 2 periods later, which is a particular impulse response, first write the above equation of evolution one period lagged:

From this, the effect of the j-th component of et−2{\displaystyle e_{t-2}} upon the i-th component of yt{\displaystyle y_{t}} is the i, j element of the matrix A2.{\displaystyle A^{2}.}

It can be seen from this induction process that any shock will have an effect on the elements of y infinitely far forward in time, although the effect will become smaller and smaller over time assuming that the AR process is stable — that is, that all the eigenvalues of the matrix A are less than 1 in absolute value.

Christopher Sims has advocated VAR models, criticizing the claims and performance of earlier modeling in macroeconomiceconometrics.[6] He recommended VAR models, which had previously appeared in time series statistics and in system identification, a statistical specialty in control theory. Sims advocated VAR models as providing a theory-free method to estimate economic relationships, thus being an alternative to the "incredible identification restrictions" in structural models.[6] VAR models are also increasingly used in health research for automatic analyses of diary data[7] or sensor data.